MECHANICAL PHILOSOPHY.-HYDROSTATICS. [SPECIFIC OHVITT. 



.-. 1 ox. avoirdupois - ^ os. troy - -911458 o*. troy 



v ~ iuvs-S "" bches - -s-iwS cubio inche8 ' 



The divisor -52746 is the troy weight in or of a cubic 

 jiH-h of water, as is obvious : it is the value of tc when 

 V-landS-l. Th.> tr,.y w.i;',it of a cubic men ot 

 water in grains U 253-1808 grains. . 



It is in general easier to ascertain the weight of a 



minute body than to measure accurately its dimensions ; 



and thus tho following is the method usually recommended 



:;nding the diameter of a small sphere of known 



specific gravity and ascertained weight in grains troy. 



Let d bo the diameter and tcthe weight in grains troy of 



the sphere, and 8 the specific gravity of the substance of 



which it is formed ; then the volume of a sphere whose 



.,eter is 1 inch being -523598 inches, we have for the 



troy weight of an equal sphere of water 



1 : : 253-1808 grains : 132 '5648 grains, 



the weight of a sphere of water of 1 inch in diameter. 



Now the volumes, and therefore the weights, of spheres 

 of the same substance being as the cubes of their dia- 

 meters, we have 



1 : d : : 132-5648 grains : 132'5648<Z S grains, 

 the weight of the small sphere of water of diameter d ; 

 therefore the weight to of the proposed sphere is 



tc= 132 5648d. S grains .' . d= -19612 Jg 



the number of grains being put for f. 



When a body is wholly immersed in a fluid, the weight 

 lost is to the whole weight as the specific gravity (S) of 

 the fluid to tho specific gravity (S') of the solid. 



Tho weight lost is the weight of the displaced fluid 

 (page 754) ; and, from the general relation WSVf, the 

 weight of bodies of the same volume are as their specific 



..s ; therefore, 

 wt. of displaced fluid : wt. of solid : : S : S' 



that is, wt lost : whole wt. : : S : S' . . . (1) 



If the same body be immersed in another fluid whoso 

 specific gravity is S , , then 



wt. lost : whole wt. : : S, : S' . . . (2) 

 Tho second and fourth terms being the same in the pro- 

 portious (1) (2), it follows that the weights lost in the two 

 fluids are as the specific gravities of those fluids ; and 

 hence may be ascertained the specific gravity of auy fluid 

 (obtainable in sufficient quantity for the experiment), 

 when the specific gravity of auy other fluid is known. 



Also, the true weight of a body may be readily deter- 

 mined, from knowing what it weighs in each of two fluids 

 of known specific gravities ; for let W be the true weight 

 and tc, tc 7 the weights in two fluids whose specific gravities 

 are S, S' : then the weights lost are W-tc and W-tc' 

 respectively. 



.-. W tc: W wf ::S : S'. 



/. g w 8'ic = 8 W Stc'. 

 .'.(8' S)W S'tc Su/. 

 8' 10 SM/ 

 .".W fyf Q~* j true weight ol 



the body. 



And thus, from having the true weight of the body, 

 and tin- weight lost in one of the fluids, wo may, by the 

 jir<i]>ortioD (1) above, find also tho specific gravity of the 

 i-.ly. 



I'.y the true or absolute weight of a body is to be 

 undcrsUKid the weight of it in a vacuum, that is, fr. . 

 fmm the presence of even tho air : tho absolute weight, 

 . U equal to the weight of the body in air 

 iiicn-axi-d by tho absolute weight of a volume of air c.|u;i 

 to that of thn 1 - ..iy ; but for bodies of small volume this 

 minute im n.-aiw is inappreciable. 



pit 1. A piece of uu iform substance, whose true 

 weight u Ct grains, is found to weigh only 48 grains 

 when immersed in distilled water : required the spociti 

 gravity (S) of the substance. 



Too weight lost u 64 48 16 grains, 



l.f- IKI 



.'. 16 : 64 : : 1 (specific gravity of water) : 1. 

 Hence the specific gravity of the body is S - 4. 



.V body woiglis 130 grains in one fluid, and 68 in 



another ; but the true weight of the body is 240 grains : 



required the relative specific gravities of tho two fluid*. 



The weights lost are 110 grains and 172 grains, and, 



aa before, these are aa the specific gravities of tho 



laid* 



8 110 11" , 



'S'~l. = 172 



If one of the fluids be air in which the body is weighed, 

 and the other water of specific gravity 1, then knowing 

 be specific gravity S of the air, we have for tin- 

 eight W of the body, that is, for its weight in vacuo, 



W ^~- - tc + (tc /) 8 + (ic tcO 8' + etc. 



= tc + (tc to 7 ) S very nearly, 



because S being a small fraction, S 1 is an equally small 

 i-action of it. 



The scales used in weighing bodies for the purpose of 

 ascertaining specific gravities are called the hydrostatic 

 Manx (Fig. 169*). The -- 



!>ody, after being weighed 

 in the air, is suspended by 

 a fine thread from one of 

 the scale-pans, and im- 

 mersed in distilled water, 

 as in the margin ; and when 

 the weights in the other 

 scale-pan bring tho scale- 

 beam into a horizontal po- 

 sition, the weight of the 

 body in the fluid will be 

 determined. 



But in what has preceded, the solid is supposed to be 

 heavier than an equal volume of tho fluid in which it is 

 immersed ; when it is lighter than tho fluid, the method 

 of proceeding is this : 



Take a body sufficiently heavy to sink the lighter body 

 with it, when both are united and immersed together in 

 the fluid. Find the weight, in the fluid, of the heavier 

 body by itself ; then the weight, in the fluid, of tho 

 united mass : this latter weight will obviously be lest 

 than the former ; as the lighter body displaces more than 

 its own weight of water, the difference will show In >w 

 much more. Call this difference between the two wi 

 to', and the true weight of the lighter body w, thn 

 weight of the fluid displaced by the lighter body will be 

 to + tc'. Consequently, tc + to 7 : tc : : S (specific gravity 

 of fluid) : S' (specific gravity of the solid). 



The weight to', by which the weight of the displaced 

 fluid exceeds that of the body, expresses the buoyancy of 

 the body, or the upward pressure upon it, in virtue of 

 which it would begin to ascend if left to itself. 



If the specific gravity of a body in the form of powder 

 U to be found, the preceding method may still be em- 

 ployed : the powder may be imbedded in wax, or some 

 such yielding material, sufficiently heavy to cause the 

 compound to sink in the fluid ; the weight in tho fluid, 

 of the wax by itself, being found, and tli.-n tin' weight 

 in the fluid, of the compound, the specific gravity of the 

 ponder, previously weighed in vacuo, will be given by 

 the foregoing formula. 



PROBLEM. If in two fluids which do not mix, a solid 

 be immersed, and rest partly in one fluid and partly in 

 tho other, to find the ratio of the t,. parts v.h.n the 

 specific gravities of the solid and the fluids are known. 



Let V be the volume immersed in the lower or heavier 

 fluid, and V the volume imiiiiTsnl in the upper <>r lighter 

 fluid ; then from the condition W VSg we necessarily 



VS + V'S' - (V + V) 8", 

 whore S', 8 are tho ppccific gravities of tho two fluids, 

 and S* the specific gravity of the solid. 



